Questions tagged [forcing]
Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).
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The proof of Lemma 10.1 in Nik’s book about forcing
I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set.
In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ...
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Laver's Proof that $V$ and $V[G]$ have the $\delta$-approximation property.
Let $V$ be a transitive model of ZFC, $\mathbb{P} \in V$ be a forcing poset such that $(|\mathbb{P}| < \delta)^{V}$ for some regular cardinal $\delta$ in $V$, and $G$ be a $V$-generic filter over $\...
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Cohen reals satisfying a formula
Consider the Cohen forcing $\mathbb{C} =Fn(\omega,2)$, the one that adds a Cohen real, and now suppose that for a Cohen real $r$ generic over $V$ we have $$V[r]\models \exists x (x \in \mathbb{R} \...
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A question on Kanamori's Book (Proposition 10.20)
I don't see a case involved in the proof of Proposition 10.20 in Kanamori's book.
Let $P$ be a separative poset with $|P|\leq|\alpha|$, for an ordinal $\alpha$. Suppose $P$ forces that there is a ...
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Forcing relation: names and check names
Let $P\in V$ be a forcing notion and let $\varphi(x,\sigma_1,\dots,\sigma_n)$ be a formula in the forcing language, where $\sigma_1,\dots,\sigma_n$ are $P$-names. Suppose there is a condition $p\in P$ ...
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A property of product forcing
I've read in an article the following statement (which is said to be a standard fact in forcing)
Given $\mathbb{P}$ a forcing notion and $G_1,G_2\subseteq\mathbb{P}$ such that $G_1\times G_2$ is $\...
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How to prove that the definitions of $MA_{\kappa}$ and $FA_{\kappa}(ccc)$ are equivalents
I'm trying to write down the steps which every basic set theory book usually just assume, which is the equivalence between the statements below (where $\Gamma$ is the family of posets which are ccc).
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About Countable Support iteration of Proper forcing notions and CH
Assume $CH$. $P$ is a countable support iteration of proper forcing notions of length $ \lt \omega_2$ and each forcing notion is forced to have size $ \le\omega_1$.
I can’t show $P$ forces $CH$. I ...
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$FA_{\kappa}(\mathbb{P}) \iff BFA_{\kappa}(\mathbb{P})$, if $\mathbb{P}$ is a $\kappa^+$-cc poset.
I'm trying to prove the question in the title, where the axioms are enunciated like
$FA_{\kappa}(\mathbb{P})$ is the assertion that for each colection $\{I_{\alpha} : \alpha < \kappa\}$ of maximal ...
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why does the union of a generic filter cannot be the zero function?
I'm trying to understand some concept of the forcing idea, but I'm having trouble with two questions.
assume that M is countable s.t. $M\models ZFC$, $\mathbb{P}$ is cohen forcing, $\mathbb{P}\in M ...
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How to more rigorously formalise “value of (Weaver's) P-name” in set theory forcing (recursion)?
Background
I have been reading some introductory material on forcing, specifically Nik Weaver's Forcing for mathematicians. What Weaver calls a “$P$-name”, I will call “Weaver's $P$-name” because it ...
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Destroying the Mahloness of $\kappa$ with a forcing of size $\kappa$ that is $\alpha$-distributive for all $\alpha<\kappa$
Exercise 21.4 of Jech's Set Theory says:
Let $\kappa$ be an inaccessible cardinal. There is a notion of forcing $(P,<)$ such that $|P| = \kappa$ and $P$ is $\alpha$-distributive for all $\alpha &...
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To study Forcing
I'm a brazilian math undergraduate student. I study at UFAM (Universidade Federal do Amazonas). My scientific initiation advisor is from the area of Functional Analysis. However, he also wanted to ...
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Further question on previous post "What Does the Forcing Relation Mean Semantically"
Now includes edits to the question to correct for the mistake in Equation (3), as identified by vsotvep
Could I ask a question relating to the last step in the elegant answer to What does the Forcing ...
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MA($\omega_1$) implies all Aronszajn $\omega_1$-trees are special
I am interested in finding proofs of the result mentioned in the title. I've heard that it can be proven in many different ways.
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Computability and Forcing
In view of the paper "Forcing As A Computational Process" by J. Hamkins, R. Miller and K. Williams, I have revised my original question, part (a) about the computability of the Forcing Truth ...
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About dense sets used in the definition of forcing equality?
In Kunen's Set Theory: An Introduction to Independence Proofs, Ch. VII, section 3, page 195, in definition 3.3 about forcing equality, it reads:
$p \Vdash^* t_1 = t_2 \text { iff } \\ \space \space (...
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Jech *Set Theory* Lemma 14.18 proof [duplicate]
Lemma $14.18$ in Jech's Set Theory states the following when expositing forcing.
If $W$ is a set of pair wise disjoint elements of a Boolean algebra $B$ and if $a_u$, $u\in W$, are elements of $V^B$, ...
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Boolean-Valued models on a proper class universe
In Jech's Set Theory he defines a Boolean-Valued model to consist of a Boolean universe $A$ and two functions of two variables with values a complete Boolean algebra $B$. The functions are $||x = y||$...
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Dense embeddings between forcing posets
I am studying Kunen's set theory book (the new version), and I am thinking about Exercise IV.4.8. This is a problem about dense embeddings between forcing posets. The relevant terminology can be found ...
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Which ordinals can be "mistaken for" $\aleph_1$?
I've just finished working my way through Weaver's proof of the consistency of the negation of the Continuum Hypothesis in his book Forcing for Mathematicians. One of the key points in this proof is ...
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The Solovay model(s) and ordinal definability
Across the literature, there are (a priori) different things that are commonly called the Solovay model. More precisely, one forces with $\text{Col}(\omega,<\kappa)$, where $\kappa$ is inaccessible,...
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Several forcing axioms imply $2^{\aleph_0 }= \aleph_2$. What about $2^{\aleph_1}$?
On the one hand, it seems intuitive that $2^{\aleph_1 }> 2^{\aleph_0}$, because $\aleph_1 > \aleph_0$. However, I also know that, like many things involving the continnum function, that's ...
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Lemma 2.20 of Kunen
I'm trying to understand the proof of a result in Kunen's Set Theory An Introduction to Independence Proofs Chapter VII, but I really can't find a way.
We want to show that if $G$ is a $\mathbb{P}$-...
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About $\lambda$-saturated ideals defined through a $\lambda$-cc forcing
In Kanamori's "The Higher Infinite" (2nd ed.) there is Lemma 17.7. When I first went through the proof it seemed obvious, but now that I'm reviewing it, I get confused.
The lemma states ...
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Does the absolute fragment of second-order logic satisfy a strong Lowenheim-Skolem property?
Let $\mathsf{SOL_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such that for every (set) forcing $\mathbb{P}$ and ...
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Countably many dense subsets in forcing
I am reading some introductions to forcing such as this one https://arxiv.org/pdf/0712.2279.pdf and they seem to take for granted that the family of dense subsets of a countable, separative, atomless ...
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What is meant by $HOD(S)$ in this paper?
Spencer Unger wrote a short paper on Solovay's Model, and in this he shows that if $S$ is the class of increasing sequences of ordinals, then the model of $\textsf{ZF+DC}$ that we are after is $HOD(S)$...
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Error in the proof of $\textit{Sheaves in Geometry and Logic}$ Lemma VI.3.8 - Cohen Poset satisfies Countable Chain Condition
Context
Chapter VI sections 1-3 in Sheaves deals with proving that there is a certain Grothendieck topos $Sh_{\neg \neg}(\mathbb{P})$ in which the continuum hypothesis is false in the internal logic.
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applying Martin's axiom
I am struggling with the following problem:
Let $A_n$ for $n \in \omega$ be sets of respective cardinality $n + 1$, and let $X = \prod A_n$. Show that Martin's axiom for $\kappa$ proves that for every ...
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Forcing Computable Functions
I was wondering about the following issue and could not see how to address it. Let $Comp_V(\omega)$ be the set of all computable funtions over $\omega$ in some universe $V$ of ZFC. Is it possible to ...
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Adding a closed unbounded set containing of only limit ordinals with a special property
The following theorem and proof are in Applications of the proper forcing axiom, the Baumgartner's paper in the book Handbook of Set-theoretic topology.
$3.6$
THEOREM. Assume PFA. Suppose that for ...
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Does Cohen forcing add Sacks real?
It is known that Sacks forcing does not add Cohen reals since it has the Laver property. I wonder if Cohen forcing can add a Sacks real. If $V$ is a model of ZFC, and $V[c]$ is the extension by adding ...
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Jech Exercise 16.4: What is $B:D$?
What is the notation $B:D$ that Jech uses in Exercise 16.4? Here's the statement of the question:
Prove $|| D\;\text{is a complete Boolean algebra}\;||_{B} = 1,$ and $D$ is isomorphic to $B\ast (D:B)$....
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Question on Jech's introduction to Iterated Forcing (What really is $P\ast Q$?)
Jech's introduction to iterated forcing is really confusing me. He starts with two-step forcing, but to be honest, even on the first page of Chapter 16, the first definition where he defines $P\ast Q$ ...
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Analytic set in forcing extension
I am trying to understand the proof of the following lemma from the paper Can the fundamental group of a space be the rationals? by Saharon Shelah.
Let $\mathcal{E}$ be an analytic equivalence ...
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Questions in Jech's presentation of the Independence of CH (p. 219)
I just had a few questions regarding Jech's proof of the Independence of CH. This is on page 219 of the most recent edition. There's just a couple lines in here I'm not quite seeing. Maybe they're ...
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What is the role of excluded middle in classic forcing arguments?
I was doing an exercise out of Jech's "Set Theory," and noticed something somewhat interesting. This is exercise 14.7 in Jech.
Let $D=\{q:q\Vdash\varphi\}$ be dense below $p$. If there ...
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Exercise 16.1 of Jech's Set Theory
Exercise 16.1 of Jech's Set Theory says:
$B(P * \dot{Q}) = B(P) * B(\dot{Q})$.
Here, $P$ and $Q$ are (forcing) posets, and $B(P)$ and $B(Q)$ are their respective Boolean completions.
Lemma 16.3 of ...
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Is the property that "a commutative domain $A$ is a PID" absolute between models of ZFC?
The property that "a commutative domain $A$ is a PID" is downward absolute. In fact, it can be written by
$$(\forall I \subseteq A \text{ ideal})\ (\exists x \in A)\ (\forall y \in A)\ [y \...
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Role of a witness in forcings that preserve a relation on the reals
I'm working through Martin Goldstern's "Tools for Your Forcing Construction" and am confused about definition 5.11.
First some notation:
$\mathbf C\subseteq{}^\omega\omega$ is some closed ...
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Unbounded set in $V[G]$ has an unbounded subset in $V$?
Suppose $\kappa$ is a cardinal preserved in the generic extension $V[G]$. Let $Y \subseteq \kappa$ be an unbounded set in $V[G]$. Does there always exist an $X \in V$ such that $X \subseteq Y$ and $X$ ...
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Proving Proposition 10.10 of Kanamori's The Higher Infinite using partial orders [duplicate]
Proposition 10.10 of Kanamori's The Higher Infinite states:
Suppose that $R$ is a [poset], $G$ is $R$-generic over $V$, and $N$ is a transitive $\in$-model of $\mathsf{ZFC}$ such that $V \subseteq N \...
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Mixtures in Boolean algebras
The following appears (with slightly different notation) as problem 1.26 in Bell's Set Theory: Boolean-Valued Models and Independence Proofs, 3rd edition:
Let $B$ be a complete Boolean algebra and $\{...
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Are there models of set theory where every real number is definable but not every set is definable.
I know that Joel David Hamkins has constructed a model of set theory where every set and hence every real number is pointwise-definable. But, is there a model of set theory where every real number is ...
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Has there been any attempt to classify the countable ordinals that correspond to models of set theory?
Let $S:=\{ \alpha\in \aleph_1 \mid \exists M, M \models ZFC \wedge M\cap \textbf{ORD} =\alpha \wedge |M|=\aleph_0 \}$. Has there been any research into the properties of S?
In particular, I'm ...
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Is the well-orderability of the reals independent from dependent choice? [duplicate]
Working in $ZF$+"There is a well-order of the real numbers", is there any known model that also satisfies $\neg DC$? I have tried to look at this in Jech's "The Axiom of Choice", ...
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Proving that the partial order for adding a dominant real is weakly homogeneous
I am trying to prove that the partial order that adds a dominant real is weakly homogeneous. This is listed as exercise IV.4.17 of Kunen's Set Theory (the "new" one).
The details are as ...
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What is the requisite knowledge in logic required to study forcing?
I have been spending the last month or so reading and doing the exercises of Chapters 1-6 of Jech's text, however I noticed a pattern in how Part I is constructed. The way it looks, of the 12 chapters ...
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Generic filters and projections
Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions and $\pi:\mathbb{P}\rightarrow\mathbb{Q}$ be a projection.
Let $G\subset\mathbb{P}$ and $H\subseteq\mathbb{Q}$ the filter generated by $\pi''G$...
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